IL NUOVO CIMENTO VoL. 32 A, N. 2 21 Marzo 1976

On a Class of Lagrangian Models for Massive and Massless Ymlg-Mills Fields.

G. C ~ c I and ]~. FERRARI (*)

Ist i tuto di ~is ica dell' Universit~ - .Pisa, I ta l ia

Ist i tuto ~Vazionale di ~'isica Nucleare - Sezione di -pisa

(ricevuto il 15 Ottobre 1975)

Summary. - - We discuss renormalization for a class of Lagrangians containing an explicit mass term for the vector field and a four-vertex for the Faddeev*Popov ghost field. The action is shown to be invariant under nonlinear gauge transformations, which are a generalization of those introduced by Becchi, Rouet and Stora. This invariance guarantees the validity of the Slawmv-Taylor identities, which are then used to discuss the renormalization of the theory. The problem of unitarity is not discussed here. Canonical and Lorentz-covariant formalism is employed.

l . - I n t r o d u c t i o n .

The description of self-interacting vector mesons (1) in terms of local fields

has always encountered a number of difficulties. To mention some of them,

the massless Yang-Mills theory is still affected by unsolved problems, e.g.

infra-red divergences and consequently the proof of unitari ty.

On the other hand, massive vector mesons can be described by making

use of the Higgs phenomenon in a theory £ la ¥ang-Mills; this approach has

the characteristic feature of requiring the spontaneous break-down of the gauge

(*) Address after November 1, 1975: Max-Planck-Institut fiir Physik und Astrophysik, Miinich, Germany. (1) A complete bibliography about the early works on the gauge fields can be found for instance in E. E. AB:ERS and B. W. L~]~: .Phys. Rep. , 9 C, No. 1 (November 1973).

151

1 5 2 G. c u R c ~ a n d R. F~RR~_R~

symmet ry (of the first kind) and the presence of some ext ra massive scalar particles. These aspects of the Higgs phenomenon are very interesting for some phenomenological problems (for instance in the unified theory of weak and electromagnetic interactions), bu t they can be drawbacks in other situa- tions. For instance, one may desire to preserve the symmet ry of the Lagran-

gian in the solution (e.g. for describing hadronic vector mesons where isotopic

spin is a good symmetry) or to consider the limit of zero mass in order to s tudy the infra-red problem of the massless Yang-Mills theory. In the presence of the Higgs phenomenon this is either impossible or presents great difficulties.

I t would be of great help, also for other problems, if one could reproduce the situation of quan tum eleetrodynamies. There one can construct a (~ mas- sive ~> QED and then consider the limit of zero mass for the theory.

In dealing with the massless Yang-Mills theory the Slavnov-Taylor identities are an impor tan t tool in order to prove renormalizabil i ty and uni tar i ty . These

identities are easily derived by making use of the invarianee of the action under the nonlinear transformations introduced by BECCI~, t~OUET and STORA (BRS transformations) (2).

The start ing point of this paper is a s traightforward generalization of the BRS transformations. We then show tha t a class of Lagrangian models for ¥ang-Mills fields yields an action invar iant under these transformations, even when mass terms are allowed; however a four-ver tex for the Faddeev-Popov ghost is needed.

Then we discuss the renormalization of these Lagrangians and show tha t the renormalized action has an invariance analogous to tha t of the unrenormalized ones. The equations suggest tha t the limit for zero mass is smooth, i.e. the Green's functions of the massive Yang-Mills theory tend to those of the mass- less ease when m goes to zero.

In this paper we do not discuss the problem of physical unitar i ty . Some equations suggest tha t iormally uni tar i ty can be proved for the massless ease just by a simple generalization of the techniques previously used by a number of authors (1,2) in similar cases. When m ~ 0 we do not know if the theory can be made physically un i ta ry ; however, for studing the infra-red problem by taking the limit of m going to zero, we should not lose the great advantage

of having Slavnov-Taylor identities also for m :~ 0. In sect. 2 we present the Lagrangiau models. They are renormalizable

just by power counting. We discuss both the massless and the massive case at the same t ime; in this way one can see the similarities of the two situations

and how the limit for zero mass works. In sect. 3 we derive the Slavnov-Taylor identities. They hold in bo th

cases (m = 0 and m ~ 0). I t is ra ther lengthy to derive them by combina-

(2) C. B~.CCHI, A. ROUET and A. STORA: Phys..Sett., $2 B, 344 (1974); Comm..Math. Phys., 42, 127 (1975).

ON A CLASS OF LAGRANGIAN MODELS FOR MASSIV]," ETC. 1 5 3

torial methods (a) or by using the equations of mot ion (% I t is be t t e r to use the action principle and to exploit the invar iance proper t ies of the La- grangian. We then derive other relations which are needed to discuss renor-

realization. In sect. 4 we write the Slavnov-Taylor identities for the one-particle

irreducible (OPI) ampli tudes. We shall use t hem for renormMization.

In sect. 5 we show tha t the theory is renormalizable by rescaling fields

and paramete rs . Thus it is easily seen t ha t the renormalized action is invar ian t

under a new set of BRS t ransformat ions .

2. - The model.

We consider the following unrenormal ized Lagrangian (~ :/: 0):

{ - - ~ * x a , , ~ + ~ a , , ( ~ o * x ~ ) ) . A , ' + g2(a--~)(~*×~o) o- + ~-m , , - - - ~ * - ~ ,

IX

where

(2.2) Gu, = ~uA~ -- OvAu + gAu x A ~ .

The covar ian t der ivat ive is

(2.3) D~ j = (~"~u + ge~kJ A,.k

We quant ize the theory b y imposing canonical commuta t i on relations. (( g h o s t , field ~ is scalar and obeys an t i commuta t ion relations.

The equat ions of mot ion are

(2.a)

(2.5)

(2.6)

The

(DUGu,) ~ + o~b, " i 1 a Ao--j. • [~cp* × $ --¢,p* × ~(p + flO,(tp* × ¢,p)]' + m~A~ = O,

m ~ 1

1 ~ 1_/~2)(¢~,x¢~)× = 0

+ l a g ( ¢P ,

1 4a g~(1 --fl~)(~* X ~ ) X ~ * --= 0.

(3) G. 'T HOOFT and M. V]~LTMA~¢: Diagrammar, CERN Report 73-9 (1973). (4) G. CURCI and R. FnRRA~I: Nuovo Cimento, 30A, 155 (1975).

154 G. euRc~ and ~. FERRARI

I f ~o* is the t t e rmi t i an conjugate of ~, then the Lagrangi~n is not Hermi t i an ,

unless fl = 0. The Lagrangian is i nva r i an t under the following t rans format ion (3) R :

(2.7) R :(v*-->-- ~v,

I R:~ - , - - ~ ,

Au, ~, m, g unchanged. The action is invar ian t under R, thus we can obtain, in general, more relat ions by taking the R- t rans fo rm of every equat ion der ived

f rom the act ion principle.

When fl = 0 the theory is t t e rmi t ian . Moreover R is a t r ans fo rma t ion on the fields only. This implies t ha t every opera tor odd under R has zero v a c u u m

expecta t ion valu% e.g.

(2.s) <0lop* ×¢p(x) A(1) ... A(n) lO } [~= o = O .

For m = 0 the Lagrangian (2.1) reduces to model of FADDEEV and P o P o v (5)

in the case fl = 1 and to t ha t of FlCADKIN and TYUTIN (e) for f l - - - - - 1.

The theory is renormalizable. This can be seen b y power counting. All the t e rms in (2.1) have dimension less than or equal to four.

3 . - S l a v n o v - T a y l o r ident i t i e s .

We derive now the identit ies t ha t we shall use later. The shor tes t way

is to consider the s y m m e t r y propert ies of the Lagrangian (2.1) under the fol-

lowing t ransformat ions :

(3.1)

8A'.(x) = ~ ( D . ~ ( x ) ) i ,

~(x) = -- -~g ~w(q~ xq~)qx),

where 8o) is a constant p a r a m e t e r which an t i commutes with ~ and ~o*.

Le t us define

1 (3.2) c = ~ , ' A , + ~ g ( 1 - 3 ) , ~ * × ~ .

(5) L. D. FADDEEV and V. N. PoPov: Phys. Lett., 25 B, 20 (1967). (e) E. S. FI~ADKIN and J. V. TYUTIN: Phys. Rev. D, 2, 2841 (1970).

155 ON A CLASS OF L A G I ¢ A N G I A ~ , MODELS FOI¢ ~[ASSIVE ETC.

Under (3.1) we have the following t ransformat ion propert ies :

( 1 , ) (3.3) ~c= ~o ~,D,,.p---~g(~ - - / ~ ) ~ , A , × , ~ + ~ g ~ ( 1 --Y')(,O* x , ~ ) × , ~ ,

(3.4) ~(D.~) = ~(~ × ~ ) = o ,

]) , ?.(¢ (3.5) ~ d~x m'A~ - - *"0 =--t~ 2o~m~g~o × ~ ) ' q ~ ; 2 ~ - -

with the help of (3.3)-(3.5) it can be easily shown tha t the first t e rm and t h a t in curly brackets in (2.1) are separa te ly invar iant . The mass pa r t is invar ian t

only if

(3.6) t im 2=- O.

This constra int is assumed from now on.

We introduce external sources in the Lagrangian

(3.7) Lf s = .P ' .Au(x) + ~*(x).~q(x) + ~q*(x).¢p(x) + K~ ' ( x ) 'Du~(x ) +

+ ½gL(x ) . (p × ~ ( x ) + R ( x ) . ( o~ 'Au (x ) + ½g(1 -- fl)¢~* ×¢~(x)) ,

where ~7", ~ and K , an t i commute among themselves and with ~v and q~*.

For la ter use we define the generat ing funct ional of the disconnected Green's functions

in the in teract ion represen ta t ion ( ~ is the in teract ing pa r t of (2.1)). Then we hay% for ins tance (7),

(3.9) <0o~dT(A, ... A .~+~ ... ~ + . q)5.~+~ ... CLs+,)10,.> =

~'+~+~W - - ( - - i)"+'(i)t 3 - - Y l " ~ * " J , ~,.+~ . . . ~+~ ~]~+~+~ . . . ~+~+t

W is invar ian t under the t ransformat ion (3.1), therefore f rom the action

principle we have

(3.10) < O o . ~ l ~ l % } = O.

(7) In order to simplify the notations we often indicate by only one letter (in brackets or subscript)isotopic, Lorentz and position (momentum) indices. Moreover all the derivatives should be understood as operating outside the T-product.

156 G. cv~c~ and ~. F:ERRARI

We use in (3.10) eqs. (3.3) and (3.4) and the equation of motion for ~0 (2.5) (in the presence of the external current) :

(3.31)

) + J,.D~,q~+ -~g~l*.(q~xc~)-~R.(--m~q~ - a'tl) I0,.> = 0 .

Equat ion (3.11) can be expressed in terms of the generat ing funct ional

(3.12) fd.x{---- ~W ~ ~W ~W ~-~(~) ~ (~) + J~(x) ~ ~K'.(x) + v'(x) ~ % ) + i

~-aR'(x) (~'(x) - 1 i ~ ) }W = 0 .

We can derive the Slavnov-Taylor identities f rom (3 .11)by taking the de- r ivat ive with respect to ~ and n t imes with respect to J and then all the sources zero. For the disconnected amplitudes we have

(3.13) <O[T(o~8~'A~(x)A(])... A(n) + ½g(] --/?)(~v* × ~)(x)A(] ).. . A(n))lo> =

<O[T(qJ*(x)Dq:(r)A(l ) ... A(r) ... A(n))[0>, I'-,1

where ~ means omitted. We shall need la ter other formal developments of eq. (3.11).

i) By taking a derivat ive with respect to ~ and put t ing V ---- 7" ---- K---- = . L = R - - - - 0 ~ we have

(3.14) <0outlT ( - a 8~'A,, (x) -4- i f dy J (y) DqJ(y) qg*(x) -- - ½ g(1 - ~)(~* x ~)(x))Io, .>,

we take the R-transform of (3.14)

----0;

(3.15)

+ ½g(1 + ,8)(~* x ~)(x))Io,.>.,

A suitable combination of (3.14) and (3.15) gives

= 0 .

(3.16) <Oout]T(--aS~A~ (x) + 2 fdy J(Y)"

• [(1 -4- fl)Dq~(y)~*(x) -- (1 - - fl)Dq~*(y)q~(x)]) [0,s>, = 0 .

ON A CLASS OF L A ( } ~ A N G I A N MO])]SI.S 1,'O1% ]~IASSIVE ETC. 157

ii) Again we differentiate (3.11) with respect to ~*(3), ~(2) and ~(1), and the~ put alt the sources to zero except J :

<0o~,]T(~ ~ A ~ (1) ~(3) ~*(2) -- a ~ A ~ (2) ~(3) ~o*(1) -- (3.17)

iJJD~(s) ~*(2) ~*(~) + ½ gll -- ~)(v* × ~)(1) v(3) ~*(~)

½ g(1 --/9)@* x ~)(2)~(3)~*(1) -F ½g@ x q)(3)T*(2)?*(1))[0,.>, = 0.

Now we consider the R-transform of (3.17) and then we take the co-ordinates of 1 and 2 to be equal and contract the isotopic-spin indices with the antisym- metric tensor. By adding the two equations we obtain

(3.18) <0o=dT(-- (a"A, × ~*)(1)~(2) + (0"A, x ~)(1 )~v*(2)--

1 2a g [(1 --/9)(~* X~) X~*(] )~(2) 4- (1 +/9)(~* X~) X ~ 0 ) ~ * ( 2 ) ] -

] 4~ g [(~ X~)(1)(~* X ~*)(2) 4- (~v* XT*)(1)(~ X ~)(2)] 4-

" i ~ •

We define the following quantity:

(3.19) H~,(y, x) -- ½ <0o.,IT[(1 -~ fl) D~¢(y) q~*(x)-- (l -- fl) D~,q~*(y) q~(x)] lO,,,>.,.

By using the equations of motion for ~ and ~v* (2.5) and (2.6) we have

(3.20) 8 ] ( [ m 2 I

H,(y , x) = ~ <0o.,] T (1 4- fl) - - --~ ¢(y)q~*(x) -J- ~g(1 --/9).

1 o • (~'A~ x q~)(y)¢*(x) - - ~ g-(1 -- fl2)(¢, x ~o) x ?(yFf*(x) - - iS(x - - y)] - -

_(,_/9)[_~ • cf (y)¢(x) 4- {g(1 4- fl)(a~'A~,x¢*)(y)~(x) 4-

' ]) ~_ ~ag2(1 _/92)(~, x¢)x~*(y)¢(x)4- i S ( x - - y ) ]0,.)j.

From (3.18) and (3.20) we obtain

] [ m s (3.21) ~-~Tn.(y, x) = 3 <0..d T[-- T [(~ + ~)~(U)~*(x)--(1--/9)~*(y)~Cx)] +

4- ~ g~(1 --/9")[(~ x ~)(y)(¢* x ~*)(x) 4- (¢* x ~*)(y)(¢ x ¢)(x)] --

i £o~ g(1--f i")[ f JDq~(q~* x q~*)(Y)~(x) 4- f JDq~*(~ x ~)(y)q~(x)]-- i(i(x--y)) lO,.}., .

158 G. CURCI and R. FERRA.RI

iii) Final ly we differentiate (3.16) with respect to J~(z) and then we t ake the divergence:

(3.22) <0..,It ( - + x)+

+ dyJ~'(y) ~j,(z)H~,(Y, ---- .

We use now (3.21) and obta in

(3.23) (Oo.tl T--eO"A,(x)O'A,(z)---ff~[(1-t-~)q~(z)~v*(x)--(1--fi)~*(z)cp(x)]+

+ ~ g~(1 - ~)[(~ x ~)(z)(~* x ~*)(x) + (~* x ~*)(z)(~ x ~)(x)] -

i

• [(1 +~)fJ1)w~*(x)~.A,,(z)--(1--B)fJ.D~*~(x)~,A.(z)])--~(x--z)}lO,.>.=O. Equa t ion (3.23) is useful m order to discuss un i ta r i ty and the . -dependence

of the physical S-matr ix .

4. - Identi t ies for OPI vert ices .

We introduce the generat ing funct ional Z of the connected Green 's func-

tions

(4.1) Z = - - i In W .

We define the mean value of the fields

~Z ~Z ~Z (4.2) a ' , ( x ) - ~J,~(x)' e ~ ( x ) - ~ * ( x ) ' e * ~ ( x ) - ~ , ( x ) "

The generat ing funct ional of the one-part icle irreducible ampl i tudes is

(4.3) . F [ a , v , c * , K , L , R ] = Z [ J , ~ 7 , ~ * , K , L , R ] - - f d x { J ~ " a ~ + c * q q + ~ q * ' c } .

W e have

(4.4) J t = - - aa-~' " = ae* ' ~1"= E "

ON A CLASS OF LAGRAI~GIAN MODELS FOR MASSIVE ETC. 1 5 ~

Late r we shall use functionals like 8F/~R. When we take the der ivat ive of this

with respect to a, c and c*, we obtain the O P I ampl i tudes with the insert ion

of the field coupled to R (in this example defined in eq. (3.2)) (8).

The ident i ty (3.12) can be expressed in t e rms o f / " only; in m o m e n t u m space

we have

3T' 3 F ~ F 3 F 8F 3 F [ m ~ ~ ] . . . . . . . . . . o~-- = 0 . (4.5) ~a~ ~K~' + ~c~ ~L ~ + ~R ~ ~c*' + o~R~

We derive f rom (4.5), as an example, an ident i ty we shall need for renor- maliz~tion. We take the der ivat ive of (4.5) with respect to o~ (9):

3~.F 3F 3I' 3~.F 3~F 3F 3I" ~ F

3°'1 ~ 3 F 31 ~ 3°-F 3~_Y 3cl ~R 3e * ~ 3R 3e~ 3c* Rlm2--°:R 3el 3e* -- O "

Then we differentiate with respect to a 2 and let all the fields a, c, c*, K, L, R be

zero:

(4.7) 3a2 3a 3e, SK @ ~a~ ~R g e I 3(~ $ - - 0 .

5 . - R e n o r m a l i z a t i o n .

The in teract ing Lagrangian in (2.1) contains four-leg vertices with a dimen- sionless coupling cons tan t and three-leg vert ices with one derivat ive. Thus the superficially d iveIgent graphs are those with

(5.1) d ---- 4 -- E > 0,

where E is the num ber of external legs.

(8) I t is easy to see by using the Legendre transforms that

~Z ~P ~Z ~/' ~P ~/~

(9) In (4.6) we use the convention that couples of variables in the same monomial and without indices have all the indices saturated.

160 o. ouRci and ~. FERRARI

We shall however consider also graphs where some of the three-leg vert ices

do not car ry m o m e n t u m . For these the superficial degree of divergence is

(5.2) d = 4 - - E - - n ,

where n is the num ber of such vertices.

Then the superficially divergent graphs are

(5.3)

a) b)

e) f) g)

d = 2 ,

d = l ,

d = O ,

is a vector line and * a ghost line. I n order to derive the S lavnov-Taylor identi t ies we used ex te rna l sources.

The renormal izat ion p rogram mus t then t ake into account the divergences in the couplings be tween sources and fields. Moreover counter t e rms involving

only sources could in principle be necessary (,o). We shall analyse these in- finities and for this it is useful to notice t ha t K~ has dimension two and q u a n t u m numbers of a ~, e~, R has dimension two and q.n. of a ~ a~, and finally L has dimension two and q.n. of c* × c*. Thus the only possible counter t e r m involving

only sources is the one propor t iona l to R . R . We shall see t h a t we do not need any counter t e r m in R ~ and t h a t all the

divergences are logari thmic. I n the present section we shall r emove only the infinities f rom the pe r tu r -

ba t ion expansion, thus we neglect finite contr ibut ions to the counter t e rms

((~ minimul renormal iza t ion ~)). The dimensional regularizat ion is unders tood

in order to handle the divergences of the theory (n). Subsequent ly a finite renormal izat ion can be per formed in order to fix

the residua of the p ropaga to r a t the physical masses.

(10) j . ZINN JUSTIN : lectures given at the International Summer Institute ]or Theoretical .Physics (Bonn, 1974). (11) G. 'T HOOFT and M. V~ZT~N: •ucl. Phys., 44B, 189 (1972).

ON A CLASS OF LAGRANGIAN MODELS FOR MASSIVE ETC.

We can write the renormalized Lagrangian in the following way:

We have neglected in (5.4) the counter terms corresponding to the graphs g) in (5.3), since they will come out to be superficially convergent. The same happens for the terms R2 in (5.5).

We use the following parametrization for the OPI amplitudes:

11 - I1 Nuovo Cimento A.

162 o. cu~c~ and ~. ~ a ~

ge~ K (5.14) ~e~ ~e~ ~L~ '

(5.15) ~e, ~e~ 3R~: -- 2 g e ~ M '

(5.16) ~a't ~e~ ~K~ -- gs~kgt'~H + "'" '

(5.~7) (0 ]T(~ × ¢~)*(~o* × ~o*)~10 } ----- i(~' t9.

In eqs. (5.8), (5.11) and (5.16) the dots indicate other invar iant ampli tudes, which are superficially convergent ; in f~ct they are multiplied by higher powers

of the external momenta and this reduces the degree of divergence.

A) Two-leg graphs. Now we can show tha t a counter t e rm in R ~ is not necessary; in fact, by succesive differentiation of (3.12) with respect to and R we obtain

~ W ~ W (5.18) ~R, ~R~ - - i ~ - - m ~ ~ -- i~,~(~ + m~l~;~) .

The OPI p~rt of the r.h.s, of eq. (5.18) is finite and thus we do not need any

counter t e rm in R ~. In order to s tudy mass counter terms and w~ve function renormalizat ion

we use (3.21) with J ~ 0. This gives

(5.19) ~ ~y~ (OIT(D~q~(y)q~*(x)(1 ~- f l ) - (1 --fl)D~q~*(y)q~(x))lO} =

2 g~ / • = (O1T(--m°q~(y)q~*(x)~ - ~ (1--fl'~)(~×~)(Y)(~*×cP*)( x ) ] - i ( ~ ( x - y ) l O }

Let us indicate with I and J the form factors of the R-t ransform of the quan-

tities appearing in eqs. (5.12) and (5.13) and moreover

(5.20) { 2 I - - ( l ~ - f l ) I + ( 1 - f l ) l ,

23 = (1 + fl) J + ( 1 - f l ) ~ .

F ro m eq. (5.19) we have

i . e .

(5.21)

p:_r;l i m~ 1 - L = - - FjI-[- g:(1--fi2)£2, 8~

ON A CLASS OF LAGICANGIAN MODELS FOR MASSIViE ETC. 163

The constant terms in (5.21) give

(5.22) ~ - 1 - - /5~).9 ,

and those proport ional to 1)2

(5.23) i----- ] - - - - g ~ ( 1 - - f i 2 ) ~ D . 8a

If m~-- - 0~ eq. (5.22) imposes tha t the mass of the ghost is zero, i.e.

~m~= 0

in (5.4). We use eq. (3.23) with J = 0 and obtain

ap p , -1 --1"71+a 1--~--g2(1--/5~)t9o~ = 0 ,

o r

(5.24) ~ r , + + 1 - ~ (1 -/5~)~9 _r, _r,, = 0 .

We obtain again two relations for the divergent pa r t of the form factors by

using eq. (5.22) :

(5.25)

and

(5.26) ] g2(j_fi2)y2] B 1=~-~

From (5.25) and (5.26) we obtain

(5.27) m-°+ ~m ~ = -~ ~m~ 28 '

moreover eq. (5.26) tells ns tha t

Z~ = 1 for /53 ---- 1 ,

i.e. the vacuum polarization tensor (self-energy) of the vector meson is transverse. Final ly we discuss the divergent par t in (5.12) and (5.13). By construc-

tion and using the R-t ransform we can easily express J and J in the form

(5.28) J = 1 ~- (1 --/5) ~, Y ---- 1 -- (1 -[-/5) e-

(5.3~)

(5.32)

and also

(5.31')

(5.32')

From

(5.33)

(at p~

(5.34)

(for fl (5.23)

(5.35)

1 ~ O. CURCI and R. F]~RRARI

Now we use the identity (4.7) to connect I with J :

and its R-transform

(5.30) ] ( ~ p 2 B - - m 2 - ~ C) = ~ p ~ D - - - - ~ - E .

We obtain the following relations:

I(m 2 - C ) = ~ J - - E ,

I B = J D

I( ,~ 2 - c ) = v3 (~ -E)

iB = ] D .

(5.23), (5.26) and (5.32') we get

3 = 1

--~ 0) and then from (5.28)

J = J - - - - 1

=0~ J = ] = l , just by R-invariance). give

i = i = 1 = D B "

Therefore we see from (5.34) and (5.35) that

Equations (5.34), (5.32) and

(5.36) ZR,-~ 1 and Zx,---- Z3/Z ~ .

B ) Three-leg vertices. We discuss the three-leg vertices by using the re- lations among the OPI amplitudes developed in sect. 4. We derive relations for the divergent parts of the amplitudes.

We differentiate (4.6)with respect to el, e~ and o* and let the external sources be zero:

(5.37) 33/~ 3~/'

~c~ 3e 1 3a 3c2 3K 33F 3 'F 3~F 3~F

3e~ 3~ 3a 3cl 3K ~ 3~ 3o 3~ 3ol 3L

~3F ~ F ~3F 32F 3c~ 3c~ ~R 3e~ 3v* + 3c~ 3c~ ~B ~c~ ~c*

- - 0 .

ON A CLASS OF LAGRA~qGIAN MOD]~LS FOR MASSIVE ETC. 165

By using the parametrizat ion in (5.12), (5.9), (5.15) and (5.14) we obtain

(5.38) M D = (G1 - - G,) I

and

(5.39) K D -= G~ I .

If fl = 0, the theory is t termit ian and

(5.40) O~ = 0,

thus from (5.38) and (5.39)

(5.42) K = Mt~= o .

We can derive the relations among the renormalization constants by using (5.38), (5.39) and (5.36):

(5.42) zL = 21/z~ and Z~, = ( ~ , / Z 2 ( 1 - - f l , ) .

~ o w we differentiate (4.6) with respect to 01, c, and as. When the ex- ternal sources are zero we obtain

(5.43) ~a3 $a2 ~a 8el ~K 3a~ 8a 8a~ ~el ~K ~a3 ~a ~Ol ~a, ~K -[-

8a~ 8R 3a3 8el 8e'* ~- 8aa 8R 8a2 ~el 8(r* ~- 0 .

We use the parametrizat ion in (5.6), (5.8), (5.13), (5.14) and (5.16) and obtain

(5.44) F I = H A ,

(5.45) G1J = H B .

We take the ratio of the two equation and then use (5.32):

(5.46) FI(~I = A / D .

This imposes

( 5 . 4 7 ) • l / Z l = 28 /Z3 .

166 o. c~n~cr and R. ~RR~RI

Then f rom (5.39), (5.44), (5.45) and (5.46) we have

(5.48) H --~ K

(therefore Zr, = Z z = 2 J Z ~ ) .

C) ~our- leg vertices. We can obtain information on the ampli tude in- volving four ghosts or two ghosts and two vector mesons, by taking the der ivat ive of (4.6) with respect to c*, two times with respect to c, and then to a t . The super-

ficially convergent graphs according to (5.1) and (5.2) can be neglected. The equat ion obtained in this way can be expressed in terms of the parameters .

We obtain the following relations:

(5.49) MG1 -- KGx - [ - HG2 = 0 ,

(5.50) a + 2b = 0,

(5.51) I a = ( H - - K ) GI,

H - - K (5.52) J l~ - - J~((~l -~- (~$) 2 (G1 -~- (~2) = 0 .

Wi th the help of (5.48) we obtain

(5.53) a = b = 0,

i .e. we do Rot need cour t e r terms for the graphs g) in (5.3), and

(5.54) Y = (Gs + G 2 ) ( M / J ) .

We use now (5.49) and (5.45) and obtain

1 (5.55)

and therefore

~ 2

2 , = ~-: (~ - p~,).

Final ly we can s t raightforwardly obtain the ver tex involving four vector mesons. We differentiate (4.6) with respect to c and three t imes with respect to a. By neglecting superficially convergent terms and inserting the parametr i - zation for the graphs, we obtain

(5.56) Z , = Z [ / Z 3 .

ON A CLASS OF L A G R A N G I A N M O D E L S FOI4 MASSIVE ETC. 167

We use the relations among the Z's to write the Lagrangian (5.4) and (5.5) in the form

1 Z1 A (5.57) ~ f , = - - - i Z 3 ~ , A , - - 8 , A , + g - 2 ~ , × A , +

z~ 1 + - - e ~ (0,A~') ~ + 2~ 8~.¢0"" 0~qo--~ g2~(O,,.p* × ~ - - ~ * × 0,~ +

1 2 ~2 2 , ~/

1 ~ 1 @ (m~-+ Sin2) { ~ A , - - ~ , q ~ * . q o } ,

1 (5.58) 2~,R=.l~.A, + ~l*.q~+q~*.~+ ~- K , . (~O,q~+ g21A, x ~ ) +

l Z R . ~-~ 21(1 + ~ L . ~ g 2 1 ¢ O × , O + Z , (o~Z.(~,.A)+ --/L),0* x ,o ) •

The three terms in curly brackets of the Lagrangian (5.57) are now separately invariant under the renormMized transformation:

SAt = a+ ~.V' + 9 ~ (A. x ~o)' ,

_ 1 2 1 (5.59) ~9 ~ = -- a~o- g~- (qo x qo) ~ ,

2 Z3

23 (a,A,), + ~ g ~ ( 1 - ~ ) ( ~ * x ~ ) , .

We can exploit this invariance to find the renormalized Slavnov-Taylor iden- tities.

We can s tudy the limit of the theory for fl = 4- 1. We shall consider only the case fl = 1, the other case can be reduced to this by an R-transformation.

For fl = 1 the equation of motion of 9 tells us tha t

~ F ~2F (5.60) ip, Se~ ~K~ -- So, ~e; "

We take the derivative with respect to a~:

(5.61) ip~' Sa~ ~o~ SK~' -- ~a'k So, ~e~ "

By using the parametrizat ion in (5.9) and (5.16) we obtain

(5.62) H =- G~ ~nd G~ = G2 ~, i.e. fl, = 1.

168 G. CU'~CI and R. F~RRARI

• R I A S S U N T 0

Si discute la rinormalizzazione per una classe di lagrangiane contenenti un termine di massa esplicito per il campo vettoriale e un vertice a quattro per il fantasma di Faddeev e Popov. Si mostra che l 'azione ~ invariante rispetto a trasformazioni non lineari di gauge, che sono una generalizzazione di quelle introdotte da Beechi, Rouet e Stora. Questa invarianza implica la validit~ delle identit~ di Slavnov e Taylor, ehe sono poi usate per discutere la rinormalizzazione della teoria. Non si discute il problema dell 'unitariet~ fisica. Si usa il formalismo canonico e eovariante seeondo Lorentz.

Pe31OMe He HOYly~IeHO.